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Solutions périodiques symétriques de l’équation de Duffing sans dissipation. (French) Zbl 0439.34033


MSC:

34C25 Periodic solutions to ordinary differential equations
34C15 Nonlinear oscillations and coupled oscillators for ordinary differential equations
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References:

[1] Bogoliubov, N. N.; Mitropolskii, Yu. A., Méthodes asymptotiques dans la théorie des oscillations non linéaires (1962), Gauthier-Villars: Gauthier-Villars Paris
[2] Cartwright, M. L.; Littlewood, J. E., On nonlinear differential equations of the second order, Ann. of Math., 48, 472-494 (1947) · Zbl 0029.12602
[3] Duffing, G., Erzwungene Schwingungen bei veränderlicher Eigenfrequenz (1918), F. Vieweg u. Sohn: F. Vieweg u. Sohn Braunschweig · JFM 46.1168.01
[4] Ehrmann, H., Über Existenzsätze für periodische Lösungen bei nichtlinearen Schwingungsdifferentialgleichungen, Z. Angew. Math. Mech., 35, 326-327 (1955) · Zbl 0065.07401
[5] Ehrmann, H., Über die Existenz der Lösungen von Randwertaufgaben bei gewöhnlichen nichtlinearen Differentialgleichungen zweiter Ordnung, Math. Ann., 134, 167-194 (1957) · Zbl 0078.07801
[6] Hale, J. K., Ordinary Differential Equations (1969), Wiley-Interscience: Wiley-Interscience New York · Zbl 0186.40901
[7] Hale, J. K.; Rodrigues, H. M., Bifurcations in the Duffing equation with independent parameters II, (Proc. Roy. Soc. Edinburgh, Sect. A, 79 (1977)), 317-326 · Zbl 0423.34055
[8] Harvey, C. A., Periodic solutions of the differential equation \(x\)″ + \(g(x) = p(t)\), Contributions to Differential Equations, 1, 425-451 (1963)
[9] Lefschetz, S., Differential Equations: Geometric Theory (1957), Wiley-Interscience: Wiley-Interscience New York · Zbl 0080.06401
[10] Loud, W. S., On periodic solutions of Duffing’s equation with damping, J. Math. Phys., 34, 173-178 (1955) · Zbl 0067.06702
[11] Loud, W. S., Nonsymmetric periodic solutions of certain second order nonlinear differential equations, J. Differential Equations, 5, 352-368 (1969) · Zbl 0169.42202
[12] Mazzanti, S., Familles de solutions périodiques symétriques d’un système différentiel à coefficients périodiques avec symétries, (Thèse (1978), Univ. Louis-Pasteur: Univ. Louis-Pasteur Strasbourg)
[13] Minorski, N., Nonlinear Oscillations (1962), Van Nostrand: Van Nostrand Princeton
[14] Morris, G. R., A differential equation for undamped forced nonlinear oscillations I, (Proc. Cambridge Philos. Soc., 51 (1955)), 297-312 · Zbl 0065.07403
[15] Nelson, E., Internal set theory: A new approach to nonstandard analysis, Bull. Amer. Math. Soc., 83, 1165-1198 (1977) · Zbl 0373.02040
[16] Schmitt, B. V., Détermination, à l’aide d’index, de solutions périodiques de l’équation de Duffing \(x\)″ + \(2x^3 = λ\) cos \(t\), (Lecture Notes in Math. No. 280 (1972), Springer-Verlag: Springer-Verlag Berlin), 330-334
[17] Schmitt, B. V., Bifurcation de certaines familles de solutions périodiques d’une équation de Duffing, (Proceedings, VIIIth Int. Conf. on Nonlin. Osc.. Proceedings, VIIIth Int. Conf. on Nonlin. Osc., Prague (1978)), 639-644 · Zbl 0572.34037
[18] Schmitt, B. V., Deux méthodes numériques simples dans le domaine des oscillations non linéaires, (Int. Series of Num. Math. No. 48 (1979), Birkhäuser: Birkhäuser Basel and Stuttgart), 145-149 · Zbl 0416.65061
[19] Schmitt, B. V.; Brzezinski, R., Localisation numérique de solutions périodiques, (Proceedings, Int. Conf. Equadiff 78. Proceedings, Int. Conf. Equadiff 78, Florence (1978)) · Zbl 0433.34034
[20] Stoker, J. J., Nonlinear Vibrations in Mechanical and Electrical Systems (1950), Wiley-Interscience: Wiley-Interscience New York · Zbl 0035.39603
[21] Turrittin, H. L.; Culmer, W. J., A peculiar periodic solution of a modified Duffing’s equation, Ann. Mat. Pura Appl., 44, 23-33 (1957), (4) · Zbl 0082.08502
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